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Professor Charles Bailyn: Okay, welcome to the second part of Astro 160. This is going to be about black holes and relativity. And just to give you kind of a preview, the whole point of black holes is that, of course, they emit no light, so you can't see them directly. And so, the question arises, "How do you know that they're there?" And the reason you can demonstrate that black holes exist is because they're in orbit around other things and you can see the motion of the other things that interact gravitationally with the black hole.

This concept should be familiar, to a certain extent, because it's exactly the same thing we've been doing for discovering exoplanets. You don't see the exoplanet directly. What happens is that there's something else that you can see that's affected by the presence of the exoplanet. So, exactly the same thing happens with black holes. And so, we're going to use the same equations, the same concepts, to explore this very different context. So, black holes can't be seen directly. And so, instead of detecting them directly, you use this combination of orbital dynamics and things like the Doppler shift to infer their presence, and more than just inferring their presence, to infer their properties.

Now, the context is more complicated. And, in particular, we're no longer going to be using Newton's laws – Newton's Law of Gravity, Newton's Laws of Motion — because there is a more comprehensive theory that replaced Newton, which is necessary to understand these things. That more complex theory is Einstein's Theory of Relativity. So, we're going to be using some relativity rather than Newtonian physics. This gets weird very fast, okay? And so, I'm not going to start there. I'm going to start with a kind of Newtonian explanation for what black holes are, we'll do that this time, and then the weirdness will start on Thursday.

So, the first concept and the easiest way, I think, to understand black holes is the concept of the escape velocity. This is a piece of high school physics. Some of you may have encountered it before. And it just means how fast you have to go to escape from the gravitational field of a given object. If you go outside and you shoot up a rocket ship or something like that, how fast do you have to shoot it up so that it doesn't fall back to the Earth? And so, you can define an escape velocity for the Earth, or for any other object for that matter, which is just how fast you have to go to escape its gravitational field.

There is an equation associated with this. It looks like this, Vescape, that's the escape velocity, 2GM / R all to the 1/2 power. And this is the speed required to escape the gravitational field of an object; supposing that the object has mass equal to M and radius equal to R. Oh, one other assumption, I'm assuming here that you're standing on the — that you start from standing on the surface of the object. If you are on the surface. Okay.

This equation should look vaguely, but not a hundred percent familiar to you, because you derived something that looked a lot like it on the second problem set, where you worked out the relationship between the semi-major axis of an orbit and the speed and object had to go in to be in that orbit, and what — the way that calculation worked out it was the velocity equals GM over the semi-major axis, a. So that 2 wasn't there in that derivation, but otherwise, the form of this is actually quite similar to that.

And so, let me explain why that is. Here's some object.

This concept should be familiar, to a certain extent, because it's exactly the same thing we've been doing for discovering exoplanets. You don't see the exoplanet directly. What happens is that there's something else that you can see that's affected by the presence of the exoplanet. So, exactly the same thing happens with black holes. And so, we're going to use the same equations, the same concepts, to explore this very different context. So, black holes can't be seen directly. And so, instead of detecting them directly, you use this combination of orbital dynamics and things like the Doppler shift to infer their presence, and more than just inferring their presence, to infer their properties.

Now, the context is more complicated. And, in particular, we're no longer going to be using Newton's laws – Newton's Law of Gravity, Newton's Laws of Motion — because there is a more comprehensive theory that replaced Newton, which is necessary to understand these things. That more complex theory is Einstein's Theory of Relativity. So, we're going to be using some relativity rather than Newtonian physics. This gets weird very fast, okay? And so, I'm not going to start there. I'm going to start with a kind of Newtonian explanation for what black holes are, we'll do that this time, and then the weirdness will start on Thursday.

So, the first concept and the easiest way, I think, to understand black holes is the concept of the escape velocity. This is a piece of high school physics. Some of you may have encountered it before. And it just means how fast you have to go to escape from the gravitational field of a given object. If you go outside and you shoot up a rocket ship or something like that, how fast do you have to shoot it up so that it doesn't fall back to the Earth? And so, you can define an escape velocity for the Earth, or for any other object for that matter, which is just how fast you have to go to escape its gravitational field.

There is an equation associated with this. It looks like this, Vescape, that's the escape velocity, 2GM / R all to the 1/2 power. And this is the speed required to escape the gravitational field of an object; supposing that the object has mass equal to M and radius equal to R. Oh, one other assumption, I'm assuming here that you're standing on the — that you start from standing on the surface of the object. If you are on the surface. Okay.

This equation should look vaguely, but not a hundred percent familiar to you, because you derived something that looked a lot like it on the second problem set, where you worked out the relationship between the semi-major axis of an orbit and the speed and object had to go in to be in that orbit, and what — the way that calculation worked out it was the velocity equals GM over the semi-major axis, a. So that 2 wasn't there in that derivation, but otherwise, the form of this is actually quite similar to that.

And so, let me explain why that is. Here's some object.

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